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From random walk trajectories to random interlacements cerny/publications/ebp.pdf · PDF filerandom walk represents a corrosive particle wandering erratically through 1. 2 Cern y,

Nov 22, 2018

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  • ENSAIOS MATEMATICOS

    200X, Volume XX, XXX

    From random walk trajectories

    to random interlacements

    Jir CernyAugusto Teixeira

    Abstract. We review and comment recent research on random inter-lacements model introduced by A.-S. Sznitman in [43]. A particular em-phasis is put on motivating the definition of the model via natural questionsconcerning geometrical/percolative properties of random walk trajectorieson finite graphs, as well as on presenting some important techniques usedin random interlacements literature in the most accessible way. This textis an expanded version of the lecture notes for the mini-course given at theXV Brazilian School of Probability in 2011.

    2000 Mathematics Subject Classification: 60G50, 60K35, 82C41, 05C80.

  • Acknowledgments

    This survey article is based on the lecture notes for the mini-course onrandom interlacements offered at the XV Brazilian School of Probability,from 31st July to 6th August 2011. We would like to thank the organizersand sponsors of the conference for providing such opportunity, speciallyClaudio Landim for the invitation. We are grateful to David Windischfor simplifying several of the arguments in these notes and to A. Drewitz,R. Misturini, B. Gois and G. Montes de Oca for reviewing earlier versionsthe text.

  • Contents

    1 Introduction 1

    2 Random walk on the torus 72.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Local entrance point . . . . . . . . . . . . . . . . . . . . . . 92.3 Local measure . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Local picture as Poisson point process . . . . . . . . . . . . 162.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.5.1 Disconnection of a discrete cylinder . . . . . . . . . 18

    3 Definition of random interlacements 203.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    4 Properties of random interlacements 274.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Translation invariance and ergodicity . . . . . . . . . . . . . 294.3 Comparison with Bernoulli percolation . . . . . . . . . . . . 324.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    5 Renormalization 375.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    6 Interlacement set 436.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    7 Locally tree-like graphs 527.1 Random interlacements on trees . . . . . . . . . . . . . . . . 527.2 Random walk on tree-like graphs . . . . . . . . . . . . . . . 55

    7.2.1 Very short introduction to random graphs . . . . . . 577.2.2 Distribution of the vacant set . . . . . . . . . . . . . 587.2.3 Random graphs with a given degree sequence. . . . . 597.2.4 The degree sequence of the vacant graph . . . . . . . 60

    7.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

  • CONTENTS

    Index 63

    Bibliography 65

  • Chapter 1

    Introduction

    These notes are based on the mini-course offered at the XV BrazilianSchool of Probability in Mambucaba in August 2011. The lectures tried tointroduce the audience to random interlacements, a dependent-percolationmodel recently introduced by A.-S. Sznitman in [43]. The emphasis wasput on motivating the definition of this model via some natural questionsabout random walks on finite graphs, explaining the difficulties that appearwhen studying the model, and presenting some of the techniques used toanalyze random interlacements. We tried to present these techniques inthe simplest possible way, sometimes at expense of generality.

    Let us start setting the stage for this review by introducing one of theproblems which motivated the definition of random interlacements. To thisend, fix a finite graph G = (V, E) with a vertex set V and an edge set E ,and denote by (Xn)n0 a simple random walk on this graph, that is theMarkovian movement of a particle on G prescribed as follows: It starts ata given (possibly random) vertex x G, X0 = x, and given the positionat time k 0, say Xk = y, its position Xk+1 at time k + 1 is uniformlychosen among all neighbors of y in G.

    Random walks on finite and infinite graphs, in particular on Zd, hasbeen subject of intense research for a long time. Currently, there is a greatdeal of studying material on this subject, see for instance the monographs[26, 27, 28, 38, 54]. Nevertheless, there are still many interesting questionsand vast areas of research which are still to be further explored.

    The question that will be of our principal interest was originally askedby M.J. Hilhorst, who proposed the random walk as a toy model for cor-rosion of materials. For sake of concreteness, take the graph G to be thed-dimensional discrete torus TdN = (Z/NZ)d which for d = 3 can be re-garded as a piece of crystalline solid. The torus is made into a graphby adding edges between two points at Euclidean distance one from eachother. Consider now a simple random walk (Xn)n0, and imagine that thisrandom walk represents a corrosive particle wandering erratically through

    1

  • 2 Cerny, Teixeira

    the crystal, while it marks all visited vertices as corroded. (The particlecan revisit corroded vertices, so its dynamics is Markovian, i.e. it is notinfluenced by its past.)

    If the time that the particle runs is short, then one intuitively expectsthat only a small part of the torus will be corroded, the crystal will beintact. On the other hand, when the running time is large, many siteswill be corroded and the connected components of non-corroded sites willbe small, the crystal will be destroyed by the corrosion, see Figure 1.1 forthe simulations. The question is how long should the particle run to destroythe crystal and how this destruction proceeds.

    Figure 1.1: A computer simulation by David Windisch of the largest com-ponent (light gray) and second largest component (dark gray) of the vacantset left by a random walk on (Z/NZ)3 after [uN3] steps, for N = 200. Thepictures correspond consecutively to u being 2.0, 2.5, 3.0, and 3.5. Ac-cording to recent simulation, the threshold of the phase transition satisfiesuc(T3 ) = 2.95 0.1.

    Remark that throughout these notes, we will not be interested in theinstant when all sites become corroded, that is in the cover time of the graphby the simple random walk. Note however that random interlacements canalso be useful when studying this problem, see the recent papers [3, 4] of

  • Random Walks and Random Interlacements 3

    D. Belius.In a more mathematical language, let us define the vacant set left by the

    random walk on the torus up to time n

    VN (n) = TdN \ {X0, X1, . . . , Xn}. (1.1)

    VN (n) is the set of non-visited sites at time n (or simply the set non-corroded sites at this time). We are interested in connectivity properties ofthe vacant set, in particular in the size of its largest connected component.

    We will see later that the right way to scale n with N is n = uNd foru 0 fixed. In this scaling the density of the vacant set is asymptoticallyconstant and non-trivial, that is is for every x TdN ,

    limN

    Prob[x VN (uNd)] = c(u, d) (0, 1). (1.2)

    This statement suggest to view our problem from a slightly differentperspective: as a specific site percolation model on the torus with density(roughly) c(u, d), but with spatial correlations. These correlations decayrather slowly with the distance, which makes the understanding of themodel delicate.

    At this point it is useful to recall some properties of the usual Bernoullisite percolation on the torus TdN , d 2, that is of the model where the sitesare declared open (non-corroded) and closed (corroded) independently withrespective probabilities p and 1p. This model exhibits a phase transitionat a critical value pc (0, 1). More precisely, when p < pc, the largestconnected open cluster C?max(p) is small with high probability,

    p < pc = limN

    Prob[|C?max(p)| = O(logN)] = 1, (1.3)

    and when p > pc, the largest connected open cluster is comparable withthe whole graph (it is then called giant),

    p > pc = limN

    Prob[|C?max(p)| Nd] = 1. (1.4)

    Much more is known about this phase transition, at least when d is large[9, 10]. A similar phase transition occurs on other (sequences of) finitegraphs, in particular on large complete graph, where it was discovered (forthe edge percolation) in the celebrated paper of Erdos and Renyi [18].

    Coming back to our random walk problem, we may now refine our ques-tions: Does a similar phase transition occur there. Is there a critical valueuc = uc(Td ) such that, using Cmax(u,N) to denote the largest connectedcomponent of the vacant set on TdN at time uNd,

    u < uc = limN

    Prob[|Cmax(u,N)| Nd] = 1,

    u > uc = limN

    Prob[|Cmax(u,N)| = o(Nd)] = 1 ?(1.5)

  • 4 Cerny, Teixeira

    At the time of writing of these notes, we have only partial answers toto this question (see Chapter 2 below). It is however believed that thisphase transition occurs for the torus in any dimension d 3. This beliefis supported by simulations, cf. Figure 1.1.

    Remark 1.1. It is straightforward to see that such phase transition does notoccur for d {1, 2}. When d = 1, the vacant set at time n is a segment oflength roughly (N (n1/2)) 0, where is a random variable distributedas the size of the range of a Brownian motion at time 1. Therefore, thescaling n = uNd = uN in (1.2) is not interesting. More importantly, itfollows from this fact that no sharp phase transition occurs even on thecorrected scaling, n = uN2. The situation in d = 2 is more complicated,but the fragmentation is qualitatively different from the case d 3.

    The phase transition for the Bernoulli percolation on th